Abstract
We discuss for which smooth global embeddings of a metric into a Minkowskian spacetime the Hawking into Unruh mapping takes place. There is a series of examples of global embeddings into the Minkowskian spacetime (GEMS) with such mapping for physically interesting metrics. These examples use Fronsdal-type embeddings for which timelines are hyperbolas. In the present work we show that for some new embeddings (non Fronsdal-type) of the Schwarzschild and Reissner-Nordstrom metrics there is no mapping. We give also the examples of hyperbolic and non hyperbolic type embeddings for the de Sitter metric for which there is no mapping. For the Minkowski metric where there is no Hawking radiation we consider a non trivial embedding with hyperbolic timelines, hence in the ambient space the Unruh effect takes place, and it follows that there is no mapping too. The considered examples show that the meaning of the Hawking into Unruh mapping for global embeddings remains still insufficiently clear and requires further investigations.
Highlights
The Hawking radiation can be interpreted as the Unruh effect (the Hawking into Unruh mapping, named global embeddings into the Minkowskian spacetime (GEMS) approach)
We discuss for which smooth global embeddings of a metric into a Minkowskian spacetime the Hawking into Unruh mapping takes place
There is a series of examples of global embeddings into the Minkowskian spacetime (GEMS) with such mapping for physically interesting metrics
Summary
At the time when the main article [4] on the subject was written, only three embeddings of the Schwarzschild metric in a flat 6-dimensional space were known: those of Kasner [25], Fronsdal [5] and Fujitani-Ikeda-Matsumoto [26]. It is easy to see from (2.3) that at r → ∞ the radius of the circumference tends to zero (at a fixed angular speed), and the detector rests in a selected coordinate system of the ambient space, so there must be no Unruh effect in this limit. This situation is due to the fact that the embedding (2.3) is asymptotically flat, i.e. the corresponding surface at r → ∞ tends to a plane (unlike the embeddings (1.2), (2.1), (2.2)). We conclude that for all the three new 6-dimensional embeddings of the Schwarzschild metric there is no Hawking into Unruh mapping
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